Dear Readers,

See www.dialectics.org
: division by 0 [renamed ‘empty zero’]
is readily performed, and yields a finitary, ‘quanto-qualitative’ value [called
‘full zero’], in the axiomatic system of the ‘Mu’ dialectical arithmetic. This arithmetic also enables a fully
arithmetical, fully algorithmic, fully ideographical expression of “dimensional
analysis”, instead of the historically retrograde, “syncopated”-algebraic form
presently prevalent, e.g., (1)[gm.], (2)[cm.], (3)[sec.], etc.

In this ‘Mu’ dialectical arithmetic, ‘quanto-qualitative’
subscripts/denominators behave analogously to the conventional superscripts
extracted by logarithm functions:
script-level multiplications translate to subscript-level,
non-amalgamative additions; script-level divisions translate to subscript-level
subtractions.

Dynamical system state variable and control parameter
‘arithmetical quantifiers’ can be expressed in forms fully ‘‘‘qualified’’’ by
‘metrical arithmetical qualifier meta-numerals’, and by ‘ontological
arithmetical qualifier meta-numerals’, in this ‘Mu’ system of arithmetic.

As a consequence, dynamical “singularities”, arising in
finite time, resulting from division by zero in the dynamical differential
equation itself, and/or in its solution-function, can be ‘semantified’ --
rendered in their real, Gödelian-like meaning -- in a way which naturally
avoids ‘infinity residuals’, i.e., the ‘infinitely erroneous’ values that
result from interpreting “singularities” as signifying physical
“purely”-quantitative infinities.

Dynamical singularities typically signify a change in the
physical ontology that the dynamical equation models -- a change that goes
beyond what the language and ‘‘‘ontological commitments’’’ of the model
specification, implicit or explicit, can encompass. That is, they signify an ‘ontological
revolution’.

Thus, the arising of the ‘full zero’ value ‘meta-numeral’ is
typically a sign-post that the dynamical trajectory has reached a point where
such ontological incompleteness of the model specification becomes explicitly
manifest.

‘Arithmetical qualifiers’ are not a new ‘ideo-phenomenon’ in
the history of human cognitions regarding numbers, or «arithmoi». In the prelude to cuneiform in ancient
Mesopotamia, ideograms representing both ‘ontological qualifiers’ and
‘metrological qualifiers’, as well as their ‘‘‘quantifiers’’’, eventually
emerged, as documented in the work of Dr. Denise Schmandt-Besserat.

The circa 250 C.E. proto-algebraic text by Diophantus of
Alexandria, entitled “«Arithmetiké»”,
which pioneered the movement toward symbolic or ideographical algebra, featured
a syncopated proto-symbol, ‘M^o’, short for «Monad», or “unit”, which denoted a
qualitative unit of a given kind of object, or, indifferently, of a
metrological unit -- i.e., of a “qualitative unit” in either case.

In Western mathematics after the Renaissance, the presence
of ‘arithmetical qualifier meta-numerals’ in mathematical expressions entered a
long eclipse and elision, in favor of “purely quantitative” ideographies --
except for the unrecognized presence of ‘metrological unit qualifiers’ in
“syncopated” form, e.g., units of measurement like “sec.”, “gm.”, and “cm.”, when
expressing “physical quantities”.

Later, an immanent re-emergence of a new kind of ‘arithmetical
qualifiers’ ensued, with the discovery that the square root of negative unity
could be represented by the so-called “imaginary unit”, i, with rich
mathematical consequences, and with the subsequent development of the “hypernumbers”
involved in, e.g., the Hamilton Quaternions, the Cayley/Graves
Octonions/Octaves, and the “Grassmann [hyper]numbers”, all of which involve
“qualitative units”. “Vector” symbols
also represent something beyond the “purely quantitative”, combining “scalar”
quantity with ‘directionality’, or orientation.

Likewise, the recent development of set theory, and of the
various “orders” of predicate calculus in mathematical logic, all involve the
ideographical symbolization of ‘‘‘idea-objects’’’ that are not “pure
quantities”, e.g., that represent “qualities”.

For further details, see:
http://www.dialectics.org/dialectics/Applications.html
,

starting with --

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_2_of_7,19APR2015.jpg

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_3_of_7,19APR2015.jpg

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_4_of_7,19APR2015.jpg

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_5_of_7,19APR2015.jpg

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_7_of_8,22APR2015.jpg

http://www.dialectics.org/dialectics/Applications_files/Glossary,E._D._Notation_Definition,'Full_Zero'_Sign,Sheet_8_of_8,22APR2015.jpg

Regards,

Miguel

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